A generalization of supplemented modules

Abstract

Let M be a left module over a ring R and I an ideal of R. M is called an I-supplemented module (finitely I-supplemented module) if for every submodule (finitely generated submodule ) X of M, there is a submodule Y of M such that $X + Y = M$, $X \cap Y \subseteq IY$ and $X \cap Y$ is PSD in Y. This definition generalizes supplemented modules and $\delta$-supplemented modules. We characterize I-semiregular, I-semiperfect and I-perfect rings which are defined by Yousif and Zhou [12] using I-supplemented modules. Some well known results are obtained as corollaries.